Contents

Idea

Higher category theory is the generalization of category theory to a context where there are not only morphisms between objects, but generally k-morphisms between (k−1)(k-1)-morphisms, for all k∈ℕk \in \mathbb{N}.

In contrast to that, a combinatorial or algebraic model for a directed space in which the 1-dimensional paths may not all be reversible is an (∞,1)-category: it still has cells of arbitrary dimension, but only those of dimension greater than 1 are guaranteed to be reversible.

Often it is convenient in practice to consider the case where the possible dimension nn of non-trivial cells is finite. This is familiar from how a topological space that happens to have vanishing homotopy groups in dimension above some nn – a homotopy n-type – is modeled by an n-groupoid. A fully directed version of this is an n-category, which is short for (n,n)-category: non-trivial cells up to and including dimension nn, and all of them allowed to be non-reversible. Actually, it is possible to go on to an (n,n+1)(n,n+1)-category, or (n+1)(n+1)-poset; you can either consider than the nn-cells are ordered, or else consider that there are irreversible (n+1)(n+1)-cells which are indistinguishable. (Reversible indistinguishable (n+1)(n+1)-cells are all identities and so might as well not exist.)

For low nn very explicit algebraic models for nn-categories are available but in their full generality become quickly rather untractable as nn increases: the series starts with bicategory, tricategory and tetracategory. While bicategories have found plenty of applications, already the axioms of tricategories are rather involved and their theory mainly serves to produce the statement that there is a good semi-strictifications of tricategories called Gray-categories.

Indeed, there are many strictified models for higher categories: combinatorial or algebraic models that sacrifice full generality for a better concrete control. Notably there is a useful combinatorial/algebraic model for strict ω-categories which, while falling short, already goes a long way towards describing general higher categorical structures. In fact, by Simpson's conjecture every ω-category is equivalent to one that looks like a strict ω-category except for possibly having weak unit laws.

The challenge of describing fully general ∞-categories/ω-categories is to achieve a combinatorial or algebraic control of all the higher composition rules of higher cells. One can distinguish roughly two orthogonal approaches to dealing with the problem:

The basic example for such “existence laws” is the Kan-filler condition that characterizes simplicial sets that are Kan complexes, the models for (∞,0)-categories. More general higher categories are obtained by relaxing the Kan condition in just the right way. For instance by simply restricting the Kan-condition to just a certain sub-set of all cells yields the definition of simplicial sets that are called quasi-categories. These model (∞,1)-categories.

One expects that every algebraic definition of higher categories admits a construction called a nerve that maps it to a simplicial set that would constitute the corresponding geometric model.

Another approach to handle the geometric definition of higher categories is a recursive one that uses nn-fold simplicial sets. This is based on the notion of complete Segal space, which is essentially a variation of the concept of quasi-category. Its advantage is that its definition may be applied recursively to yield the notion of n-fold complete Segal spaces. These model (∞,n)-categories for finite nn.